
the list of hypergeometric equations solvable in the Liouvillian sense is completely known
(Scwharz–Kimura's list consisting of the fifteen families [1]}, see also [2]});

the Bessel equation is solvable in the Liouvillian sense if and only if its parameter is a half-integer
(see [3]} or [2]});

the equation \(y^{\prime \prime }=(az^2+bz+c)y\) , \(a\ne 0\) , is solvable in the Liouvillian sense if and only if \((4ac-b^2)/4a^{3/2}\) 
is an odd integer (see [5]});

more generally, an equation \(y^{\prime \prime }=P(z)y\)  with a polynomial \(P\) , is not solvable in the Liouvillian sense
if \(\deg P\)  is odd (see [6]}) whereas in the \((2n+1)\) -dimensional space of such equations with \(\deg P=2n\) ,
equations solvable in the Liouvillian sense form a union of countable number of algebraic varieties of dimension
\(n+1\)  each (see [7]}, [8]}).

